The two-dimensional heat equation with a random potential that is correlated in time
A talk in the SPDEvent series by
Sotirios Kotitsas from University of Warwick
| Abstract: | We consider the PDE: $\partial_t u(t,x) = \frac{1}{2} \Delta u(t,x) + \beta u(t,x) V(t,x)$ in the critical dimension $d = 2$ where $V$ is a Gaussian random potential and $\beta$ is the noise strength. We will focus in the case where the potential is not white in time and we will study the large scale fluctuations of the solution. This case was considered before in $d \geq 3$ and for $\beta$ small enough. It was proved that the fluctuations converge to the Edwards-Wilkinson limit with a nontrivial effective diffusivity and an effective variance. We prove that this result can be extended to $d = 2$. In particular we show that after tuning $\beta$ accordingly and re-normalizing the large scale fluctuations of the solution, they converge in distribution to the Edwards-Wilkinson model with an explicit effective variance but with a trivial effective diffusivity. We able to prove this for all $\beta$ below a critical value, after which the effective variance is infinite. Our main tools is the Feynman-Kac formula and a fine analysis of a specific Markov chain on the space of paths that was first introduced in $d \geq 3$ for the same problem. |